We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V₀ is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as in computing the period of a Riesz wavelet, answering in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the quasi-affine structure.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-8,
author = {Marcin Bownik and Eric Weber},
title = {Affine frames, GMRA's, and the canonical dual},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {453-479},
zbl = {1063.42023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-8}
}
Marcin Bownik; Eric Weber. Affine frames, GMRA's, and the canonical dual. Studia Mathematica, Tome 157 (2003) pp. 453-479. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-8/